I am very confused about Dickey-Fuller Unit Root Test.

According to my understanding; able to test existince of an unit root, exact data generating mechanism(not the parameter values of course, but the form of the model) be known or we must have a well approximation to the exact DGM.

After testing the unit root; If required, a suitable transformation is applied to time series. And then we try to adopt a model to the transformed(or not transformed but stationary) time series.

Let assume we try to fit an ARIMA model.

I am confused with that. In the first step(determining the model has unit root or not), we already have to choose a good model. So why don't we use the assumed model in the first step?

If we select an entirely different model in the second stage from the model used(in the Dickey Fuller Test), can be the results of the Dickey Fuller test wrong?

I live a paradox in that test. I will be very glad for any help. Thanks a lot.


Dickey-Fuller tests whether your data is $I(1)$, which is one possible source of nonstationarity. If you cannot reject the null, which is $\beta$ equal 1 in the following,

$$ x_t = \beta x_{t-1} + \epsilon $$

then any ARMA model won't make sense. So, to answer your question, no, you do not choose a model by performing DF test, you narrow down the class of models you could use for that type of data.

The popular implementations of ARIMA (notice the $I$ here) would do the DF (or ADF) test for you, and that determines the d in ARIMA(p,d,q).

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